Measuring psi(3770) -->rho pi in e+e- experiment

In S- and D-wave mixing scheme, the branching ratio of psi(3770) -->rho pi is estimated. Together with the continuum cross section of rho pi estimated by form factor, the observed cross section of rho pi production at psi(3770) in e+e- experiment is calculated taking into account the interference effect between the resonance and continuum amplitudes and the initial state radiative correction. The behavior of the cross section reveals that the disappearance of rho pi signal just indicates the existence of the corresponding branching ratio B(psi(3770) -->rho pi) at the order of 10 to the minus fourth.


Introduction
The lowest charmonium resonance above the charmed particle production threshold is ψ(3770) (shortened as ψ ′′ ) which provides a rich source of D 0 D 0 and D + D − pairs, as anticipated theoretically [1]. However, non-DD (non-charmed f inal state) decay of ψ ′′ was studied theoretically and searched experimentally almost two decades ago. The OZI violation mechanism [2] was utilized to understand the possibility of non-DD decay of ψ ′′ [3], and experimental investigations involving noncharmed decay modes could be found in Ref. [4].
It is known that at √ s = M ψ ′′ , the total continuum cross section, which is 13 nb, is larger than that of resonance. Due to the OZI suppression, the total cross section of non-DD decay from the resonance is much smaller than that from the continuum. For an individual exclusive mode, the contribution from the continuum process may be larger than or comparable with that from the resonance decay. For the ρπ mode, the cross section of the resonance decay is more than three orders of magnitude smaller than the total continuum cross section, thus the contribution from the continuum and the corresponding interference effect must be studied carefully and taken into account in case of significant modification of the experimentally observed cross section.
In the following sections, the Born order cross sections from the continuum and the resonance decays are given by virtue of the form factor and the S-and D-wave mixing model, then the experimental observable is calculated taking into account the radiative correction and experimental conditions. Finally the dependence of the observed ρπ cross section on the phase between the OZI suppressed strong decay amplitude and the electromagnetic decay amplitude is discussed.

Born order cross section of ρπ
In e + e − annihilation experiment at the charmonium resonance ψ ′′ , there are three amplitudes responsible for ρ 0 π 0 final state 2 : the continuum one-photon annihilation amplitude a c , the electromagnetic decay amplitude of the resonance a γ and the OZI suppressed strong decay amplitude of the resonance a 3g [11]: A ρ 0 π 0 (s) = a 3g (s) + a γ (s) + a c (s) .
As to electromagnetic interaction, the a c and a γ are related to the ρπ form factor: and a γ (s) = B(s) · F ρ 0 π 0 (s) , with the notation 2 Generally for certain final state f , three amplitudes describe the following three processes: The first two processes are called resonance processes, and are denoted together as ψ ′ , ψ ′′ → f for short, while the third one is called continuum process and denoted as e + e − → f for short.
where α is the QED fine structure constant, Γ t and Γ ee are the total width and e + e − partial width of ψ ′′ . The strong decay amplitude can be parametrized in terms of its relative phase (φ) and relative strength (C) to the electromagnetic decay amplitude: where C is taken to be real.
so the total ρ 0 π 0 cross section at Born order is where q ρ 0 π 0 is the three momentum of ρ 0 or π 0 in the final state.
Since there is no experimental information on ρπ cross section for the continuum process at resonance peak, the ωπ 0 form factor is used for estimation. According to the SU(3) symmetry [13], This is in good agreement with the model dependent calculation in Ref. [15] F ωπ 0 (s) where f π is the pion decay constant, or by using the ωπ 0 form factor at Q 2 = 0 from the crossed channel decay ω → γπ 0 .
With the form factor in Eq. (6), the Born order continuum cross section of ρπ production at ψ ′′ resonance peak is 3 σ Born e + e − →ρπ = 4.4 pb .
For the resonance part, ee is the e + e − partial width without vacuum polarization correction [16]. Starting from Eq. (1), it can be acquired Since there could be an unknown phase between ρ 0 π 0 |2 3 S 1 and ρ 0 π 0 |1 3 D 1 , or equivalently a phase (denoted as α) between ρ 0 π 0 |2 3 S 1 and [12], and ρ 0 π 0 |2 3 S 1 in Ref. [7], for θ = 12 • , which corresponds to the variation of α from 0 • to 180 • . So the relation between C and φ could be derived from Eq. (7). For a given B ψ ′′ →ρπ , according to Eqs. (3) and (4), the observed cross section depends on the interference pattern between the continuum onephoton amplitude and the ψ ′′ decay amplitude. In case of φ = ±90 • , the maximum constructive or destructive interference between a 3g and a c happens at the resonance peak; while φ = 0 • or 180 • leads to constructive or destructive interference between a 3g and a γ .

Observed cross section of ρπ
Due to the rapidly varying Breit-Wigner formula and the ρπ form factor as the center of mass energy changes, the observed cross section depends strongly on the initial state radiative correction which reduces the center of mass energy, and the invariant mass cut (m cut ) which removes the events produced by the initial state radiation. Taking these into account, the observed cross section becomes F (x, s) has been calculated to the accuracy of 0.1% [17,18,19] and Π(s) is the vacuum polarization factor [20]. It should be emphasized that the radiative correction modifies the Born order cross section in a profound way. Firstly, it shifts upward the maximum total cross section [21] to the energy M ψ ′′ + 0.75 MeV 4 . Secondly, the radiative correction changes the Born order cross section significantly. For example, if φ = −90 • and B ψ ′′ →ρπ = 4.1 × 10 −4 , σ Born ρπ = 5.6 × 10 −3 pb, after radiative correction σ obs ρπ = 0.31 pb for x m = 0.02. The dependence of the observed cross section on the invariant mass cut is illustrated in Fig. 1, for the branching ratio in Eq. (8) and φ = −90 • . It is obvious that a tighter invariant mass cut results in a smaller observed cross section. In the following analysis, x m = 0.02 is taken, which means a cut of ρπ invariant mass within 38 MeV from M ψ ′′ , or, near the midway between ψ ′ and ψ ′′ masses.
The phase φ between a 3g and a γ has significant effect on the observed cross section due to different interference patterns. Fig. 2(a) shows the observed cross sections at ψ ′′ resonance peak as functions of B ψ ′′ →ρπ with x m = 0.02 and φ = −90 • , 90 • , 0 • and 180 • , respectively. For the destructive interference between a 3g and a c (φ = −90 • ), the cross section reaches its minimum for B ψ ′′ →ρπ ≈ 4.1×10 −4 , which corresponds to the resonance cross section of 3.3 pb, but σ obs ρπ is only 0.31 pb, an order of magnitude smaller.
Above calculations of the observed cross section could be extended to other 1 − 0 − decay modes, such as K * 0 K 0 + c.c. and K * + K − + c.c., whose amplitudes are expressed as [12]: where R ≡ (a 3g + ǫ)/a 3g , with ǫ describing the SU(3) breaking effect. It is assumed that ǫ has the same phase as a 3g [22], so R is real. Using C determined from B ψ ′′ →ρπ and R = 0.775 from fitting J/ψ → 1 − 0 − decay [12], the cross section of K * 0 K 0 or K * + K − is calculated by Eq. (4) merely with the substitution of A K * 0 K 0 or A K * + K − for A ρ 0 π 0 . Their observed cross sections at ψ ′′ resonance peak as functions of B ψ ′′ →ρπ are shown in Fig. 2(b) for φ = −90 • and x m = 0.02. It could be seen that the cross section of K * 0 K 0 + c.c. is much larger than those of ρπ and K * + K − + c.c. in a wide range of the ρπ branching ratio.
Since the data at ψ ′′ resonance peak alone can not fix all parameters (C, φ, and F ρπ ) in B ψ ′′ →ρπ determination, the correct way of measuring the branching ratio is through energy scan of the resonance. Fig. 3 shows the observed ρπ cross section in the vicinity of the ψ ′′ resonance, with ρ π Figure 2. (a) Observed ρπ cross section as a function of B ψ ′′ →ρπ for different phases, and (b) observed cross sections of K * 0 K 0 +c.c., K * + K − + c.c., and ρπ as functions of B ψ ′′ →ρπ for φ = −90 • .
x m = 0.02 and the branching ratio in Eq. (8) for four values of φ: −90 • , +90 • , 0 • , and 180 • . The hatched areas are due to the variation of α. For φ = 0 • or 180 • , the maximum observed cross section is above or below the resonance mass. Only for φ = +90 • the maximum observed cross section is near the resonance peak. Here the most interesting phenomenon is, with φ = −90 • , the observed cross section reaches its minimum near the resonance peak! This phenomenon suggests that at the resonance peak the undetectable experiment cross section of ρπ just indicates the existence of the corresponding branching ratio at the order of 10 −4 .  72 3.73 3.74 3.75 3.76 3.77 3.78 3.79 3.8 3.81 3

Discussion
As shown in Fig. 3, the line shape of the ρπ cross section is sensitive to the phase φ. If the fine scan is infeasible, at least at three energy points the data must be taken in order to fix the three parameters in Eq. (3): C, φ and F ρ 0 π 0 . According to Eq. (1), with ρπ branching ratios at ψ ′ and ψ ′′ , and the magnitude of ρπ|2 3 S 1 , which is derived in Ref. [7], the relative phase α could also be determined. If only the data at ψ ′′ peak is available, a model-dependent way to determine the ρπ branching ratio is to look for more 1 − 0 − modes, such as K * 0 K 0 + c.c., K * + K − + c.c. and ωπ 0 . Notice in Eqs. (3), (5), (9) and (10), the three modes are parametrized by four parameters: C, R, φ and F ωπ 0 . With the measurement of F ωπ 0 through ωπ 0 mode, the other three parameters could be solved from Eqs. (3), (9) and (10).
Eqs. (3) and (9) indicate opposite interference patterns between a 3g and a c for ρ 0 π 0 and K * 0 K 0 . That is, if the interference between a 3g and a c for ρπ is destructive, then such interference is just constructive for K * 0 K 0 +c.c. and vice versa. In the resonance scan, if σ obs ρπ reaches its valley near ψ ′′ resonant mass, σ obs K * 0 K 0 +c.c. reaches its peak 5 . This means if the observed ρπ cross section at ψ ′′ is smaller than that at continuum, the observed K * 0 K 0 +c.c. cross section at ψ ′′ will be larger. So the measurements of K * 0 K 0 +c.c. and ρπ provide a crucial test of the interference pattern between a 3g and a c .
There are theoretical arguments in favor of the orthogonality between a 3g and a γ [23] of the charmonium decays. The phenomenological analyses for many two-body decay modes: 1 − 0 − , 0 − 0 − , 1 − 1 − , 1 + 0 − , and Nucleon anti-Nucleon on J/ψ data support this assumption [22,24]. The recent analysis of ψ ′ → 1 − 0 − decays which took into account the contribution from the continuum, found that the phase φ = −90 • could fit current available data within experimental uncertainties and φ = +90 • could be ruled out [12]. Similar analysis of ψ ′ → 0 − 0 − decays also favors the orthogonal phase [25]. For ψ ′′ , it is of great interest here to note the search of ρπ mode by MARK-III [4] at the ψ ′′ peak. The result corresponds to the upper limit of the ρπ production cross section of 6.3 pb at 90% C. L., which favors φ = −90 • than other possibilities as seen from Fig. 3. These experimental information suggests the phase φ = −90 • between a 3g and a γ be universal for all quarkonia decays.
At last, a few words about the effect of the beam energy spread ∆ for cross section measurement [14]. ψ ′′ is a relatively wide resonance, for a collider with small energy spread, such as ∆ = 1.4 MeV at BES/BEPC [26], this effect is negligible. With increasing ∆, the correction becomes larger. For example, on a collider with ∆ = 5 MeV, for φ = −90 • and x m = 0.02, the observed cross section of ρπ for B ψ ′′ →ρπ = 4.1×10 −4 is more than doubled to 0.68 pb comparing with the value without energy spread effect. 5 In Fig. 3, if the scan behavior of σ obs ρπ is similar to the curve corresponding to φ = −90 • , σ obs would be similar to the ρπ curve corresponding to φ = +90 • .

Summary
By virtue of S-and D-wave mixing model, B ψ ′′ →ρπ is estimated, together with the estimated σ e + e − →ρπ by form factor, the observed ρπ cross section at ψ ′′ in e + e − experiment, which takes into account the initial state radiative correction, has been calculated. The study shows that the disappearance of ρπ cross section at ψ ′′ peak just indicates the branching ratio of ψ ′′ → ρπ at the order of 10 −4 .
Besides the information of ψ ′′ , if the phase analyses of J/ψ and ψ ′ are also taken into consideration, it is natural to conclude that the phase φ = −90 • between a 3g and a γ is universal for all quarkonia decays.
In the forthcoming high luminosity experiments of ψ ′′ at CLEO-c [27] and BES-III [28], the property of ρπ decay and the feature of the phase are expected to be tested quantitatively.